Physics is becoming too
difficult for physicists.
— David Hilbert
(mathematician)
Simple Harmonic Oscillator
Credit: R. Nave (HyperPhysics)
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2 Particles, each moving in 1 D,
i.e. along the X axis
Then configuration space is 2
dimensional: x1, x2.
Here each particle is in a
superposition of two wave
packets, say ψ(x) and φ(x).
 e.g. they may have just gone
through a double slit.
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Particle 2 X
2-Particle wave functions
Position of each particle is
independent of the other, so we
can factorise wave function:
 Ψ(x1,x2) =
[ ψ(x1)+φ(x1) ][ ψ(x2)+φ(x2) ]
 or |Ψ = ( |ψ+|φ )( |ψ+|φ )
Particle 1 X
φ(x)
ψ(x)
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Now particle positions are
correlated.
Particles are still in a
superposition of two possible
positions…
…but position of each particle
depends on the other:
 if particle 1 went through right
slit, particle 2 went through left,
and vice versa.
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Particle 2 X
2-Particle wave functions
Wave function is composite:
 Ψ(x1,x2) =
ψ(x1) φ(x2) + φ(x1) ψ(x2)
 or |Ψ = |ψ|φ + |φ|ψ
 NB: two products!
Particle 1 X
Applications of Entanglement
► Tests
of non-locality
(Bell inequalities etc)
► Quantum cryptography
► Quantum computing
► Quantum teleportation
► Interface between
quantum and classical
worlds
Einstein, Podolsky & Rosen (1935)
“Is Quantum Mechanical Description
of reality complete?”
► Create entangled pair of particles
with correlated position and
momentum:
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Let particles separate and measure
mom of A and pos of B at the same
time:
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Measure momentum of particle A, tells
you automatically momentum of
particle B
Measure position of particle B, tells
you position of A
Apparently give x and p of both
particles (just before measurement)
Breaks uncertainty principle?
No practical use as measurements
disrupt p of A and x of B!
Bell Inequalities
Measurements at nonorthogonal angles for
local theory must satisfy
certain inequalities in
their probabilities of
results, which Bohm
states violate.
► Experiments (1980s)
confirm QM prediction,
i.e. violation of Bell
inequalities
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Qubit
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Classical computer: store data as binary digits (bits) 0 or 1.
Quantum computer: store data in 2-level systems (Qubits) with eigenstates |0
& |1 (or |g & |e for ground and excited)
 e.g. spin up & spin down; ground & 1st excited state of oscillator, etc.
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Quantum logic gates change state of qubit eigenstates.
 E.g. quantum not gate:
input out put
0 0  0 1
1 0 1 0
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If qubit is in superposition, output is entangled:
a 0  b 1  0  a 0
0 b1 1
Quantum computing
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In principle qubit-based computers allow massively parallel calculations
(each element of superposition acts as a separate process).
Current state-of-the-art: 4-qubit superconducting chip from University
of California, Santa Barbara (UCSB)
Problem is “decoherence”, i.e. effective “measurement” of quantum
state by unwanted correlation with environment.
Target for practical use: ~100 qubits (few years time?)
Aim is to produce faster algorithms than possible on classical
computers, for a given size of problem N (e.g. number of bits in input
data).
Algorithms developed as exercise in theory, e.g.
 1994: Shor’s algorithm: factorises large integers (would break many
common high-security cryptography systems). t  log(N)3 for quantum
computer, vs t  exp(log N /3) for best classical algorithm.
 1996: Grover’s algorithm: database search t  N1/2 quantum vs t  N
classical
Quantum cryptography
G. Von Asche / CUP
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To send secure message from A to B, even if message is intercepted by
evesdroppers:
Choose private key (very long binary number: for complete security, needs to
be as long as the message.)
Use to encode message
Send message to B
B must get key without any possibility of evesdropping.
Quantum Key Distribution (QKD) allows this, or rather, allows Eve to be
unambiguously detected.
Quantum Key Distribution
Key idea: if Eve reads the key, she
makes a measurement and hence
disturbs the message. Hence even if
she forwards message on, her
interference can be detected.
► Encode and publically transmit
message only when key
transmission has been verified.
► Quantum transmission uses photon
polarization state to encode 0 or 1,
e.g.
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Digit
0
1
BB84 Protocol:
1.
2.
3.
4.
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“Q” states
“U” states
6.
Randomly generate digits of key
Randomly choose to encode each
digit using either Q or U states, &
transmit
Bob randomly chooses Q or U states
to decode. If he gets it wrong, gets
random answer as U states are
50:50 superpositions of Q states
and vice-versa
Alice & Bob exchange list of choice
of Q vs U states (via public
channel).
Keep only cases where they
randomly made the same choice (in
which case the Bob should get the
digit Alice sent, barring
interference).
Check subsample of digits for
interference (Eve or bad
transmission).
QKD details
► What
if Eve makes a copy and “measures”
that, while forwarding original?
 Not possible: “Quantum no-cloning theorem”
shows that it is impossible to make a perfect
copy of an arbitrary unknown quantum state.
► Can
you really do this?
 YES! There are now several commercial systems
►Photons
transmitted by optical fibre or free-space
transmission
►Transmission lengths currently > 100 km
A Quantum object you can see!
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Mechanical resonator with a
fundamental frequency of 6 GHz
 Expected to be in ground state
for T < 0.1 K
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Quantum properties
demonstrated by electrical
coupling to a qubit.
 Ground and first excited states
 Superposition of the two
 Exchange of quantum with qubit
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Andrew Cleland & John
Martinis’s team at UCSB
(O’Connell et al, Nature 2010)
Recognized as “Breakthrough of
2010” by Science magazine
Dilatational resonator
(Quantum Drum, Q=260)
Piezo-electric slab (~ 1014 atoms) driven/measured by capacitor plates:
2fr 1/(LmCs), Cs-1= Cm-1 +C0-1
AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967
Coupled qubit–resonator.
► Qubit
has
tunable energy
difference: vary
around energy
quantum of
oscillator.
► Repeat ~1000
measurements
to calculate
probability of
excited state (Pe)
AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967
Qubit–resonator swap oscillations
Theory
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Experiment
Excite Qubit with microwave X pulse.
Turn on interaction with resonator for time  by tuning qubit energy to
close to resonator frequency (offset = )
Measure Pe vs (,): the quantum is shunted between qubit &
resonator and back: |Ψ = cos(t/ph)|e|0 + sin(t/ph)|g|1
After ph quantum is in resonator; after ph/2 we have entangled state.
AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967